Internal problem ID [10997]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 163.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {\left (x^{2}+a \right ) y^{\prime \prime }+2 b x y^{\prime }+2 \left (b -1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 41
dsolve((x^2+a)*diff(y(x),x$2)+2*b*x*diff(y(x),x)+2*(b-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (\frac {x^{2}+a}{a}\right )^{-b +1}+c_{2} x \operatorname {hypergeom}\left (\left [1, b -\frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {x^{2}}{a}\right ) \]
✓ Solution by Mathematica
Time used: 0.426 (sec). Leaf size: 64
DSolve[(x^2+a)*y''[x]+2*b*x*y'[x]+2*(b-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (a+x^2\right ) \left (\frac {c_2 x \left (\frac {a+x^2}{a}\right )^{-b} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},2-b,\frac {3}{2},-\frac {x^2}{a}\right )}{a^2}+c_1 \left (a+x^2\right )^{-b}\right ) \]